Integrand size = 18, antiderivative size = 18 \[ \int x^m \cosh ^2(a+b x) \coth (a+b x) \, dx=\frac {2^{-3-m} e^{2 a} x^m (-b x)^{-m} \Gamma (1+m,-2 b x)}{b}+\frac {2^{-3-m} e^{-2 a} x^m (b x)^{-m} \Gamma (1+m,2 b x)}{b}+\text {Int}\left (x^m \coth (a+b x),x\right ) \]
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Not integrable
Time = 0.10 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int x^m \cosh ^2(a+b x) \coth (a+b x) \, dx=\int x^m \cosh ^2(a+b x) \coth (a+b x) \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int x^m \coth (a+b x) \, dx+\int x^m \cosh (a+b x) \sinh (a+b x) \, dx \\ & = \int x^m \coth (a+b x) \, dx+\int \frac {1}{2} x^m \sinh (2 a+2 b x) \, dx \\ & = \frac {1}{2} \int x^m \sinh (2 a+2 b x) \, dx+\int x^m \coth (a+b x) \, dx \\ & = \frac {1}{4} \int e^{-i (2 i a+2 i b x)} x^m \, dx-\frac {1}{4} \int e^{i (2 i a+2 i b x)} x^m \, dx+\int x^m \coth (a+b x) \, dx \\ & = \frac {2^{-3-m} e^{2 a} x^m (-b x)^{-m} \Gamma (1+m,-2 b x)}{b}+\frac {2^{-3-m} e^{-2 a} x^m (b x)^{-m} \Gamma (1+m,2 b x)}{b}+\int x^m \coth (a+b x) \, dx \\ \end{align*}
Not integrable
Time = 16.83 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int x^m \cosh ^2(a+b x) \coth (a+b x) \, dx=\int x^m \cosh ^2(a+b x) \coth (a+b x) \, dx \]
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Not integrable
Time = 0.28 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00
\[\int x^{m} \cosh \left (b x +a \right )^{3} \operatorname {csch}\left (b x +a \right )d x\]
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Not integrable
Time = 0.26 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int x^m \cosh ^2(a+b x) \coth (a+b x) \, dx=\int { x^{m} \cosh \left (b x + a\right )^{3} \operatorname {csch}\left (b x + a\right ) \,d x } \]
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Timed out. \[ \int x^m \cosh ^2(a+b x) \coth (a+b x) \, dx=\text {Timed out} \]
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Not integrable
Time = 0.42 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int x^m \cosh ^2(a+b x) \coth (a+b x) \, dx=\int { x^{m} \cosh \left (b x + a\right )^{3} \operatorname {csch}\left (b x + a\right ) \,d x } \]
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Not integrable
Time = 0.29 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int x^m \cosh ^2(a+b x) \coth (a+b x) \, dx=\int { x^{m} \cosh \left (b x + a\right )^{3} \operatorname {csch}\left (b x + a\right ) \,d x } \]
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Not integrable
Time = 2.38 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.22 \[ \int x^m \cosh ^2(a+b x) \coth (a+b x) \, dx=\int \frac {x^m\,{\mathrm {cosh}\left (a+b\,x\right )}^3}{\mathrm {sinh}\left (a+b\,x\right )} \,d x \]
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